A new idea of describing ignition delay time τ for hydrocarbon fuels has been proposed, in which ignition process is divided into two phases, H₂O₂ loop induction time τ₁ and H₂O₂ loop dominant time τ₂, whether it induces a cool flame with low initial temperatures or does not with high initial temperatures. A new idea for predicting ignition timing in the cylinder with high accuracies has been proposed, in which the integral of 1/ τ₁ predicts the end of τ ₁, and then, the integral of 1/ τ₂ starts to predict the ignition timing. Ignition delay time equations for a premium-gasoline surrogate fuel have been developed, which can reproduce the temperature-, pressure-, and equivalence ratio-dependences of τ₁ and τ produced using a detailed reaction mechanism. H₂O₂ loop induction time with low initial temperatures τ_{1_low}, H₂O₂ loop induction time with high initial temperatures τ_{1_high}, and ignition delay time with high initial temperatures τhigh have been expressed using Arrhenius equations. H₂O₂ loop induction time τ₁ and ignition delay time τ have been expressed as τ ₁ = τ_{1_low} + A(τ_{1_high} – τ_{1_low} ) and τ = C{τ_{1_low} + B(τ_high – τ_{1_low} )}, respectively. Equations for A and B have been developed using Gompertz functions to express τ₁ and τ with middle initial temperatures, respectively. An equation for C has been developed using a Gompertz function to express τ with very high initial temperatures.